From the looks of these signals, your power supply does seem to be producing classic switch-mode noise. Some cheap switch-mode power supplies will notably produce a great deal more noise as the current draw increases.
If I’m reading the display right, it’s set to 2V/division and by the looks of the noise its pk-to-pk is close to 2V at times. While I don’t think this is the actual cause of the enumerated problem, it is cause for concern, especially for high-speed 3.3V SPI.
Those - half state waveforms might indicate a weak gate or a ground-ref problem.
in any case, the low-hanging fruit here is:
Use a battery for your power supply when checking timings and bit-bashing data in and out of SPI. It’ll save you some headaches.
Make sure you have a decent ground here. You can check this using a sensitive DVM placed between the DUT ground and the test equipment ground. Any voltage differential is bad news. (I once found a 120VAC difference this way - someone reverse-wired a strip where a CP/M multiuser box was blown sky-high after plugging in an RS232 cable from a Televideo 920 in another room.)